ARIMAX/SARIMAX/VAR
Multi-Variable Dataset, Model Fitting, Diagnostics, Forcasting, and Validation
Literature Review
We will conduct a brief analysis on the topics of SARIMAX, ARIMAX, and VAR models in the context of monetary policy and treasury markets. Specifically, we will explore the relationships between inflation, the Federal Funds Rate, 30-Year Treasury Yield, and other related variables.
SARIMAX Model for Inflation: The Seasonal Autoregressive Integrated Moving Average with Exogenous Variables model is a powerful tool for time series analysis that incorporates external or exogenous variables into the traditional ARIMA framework. In the context of inflation forecasting, the SARIMAX model allows us to consider how variables like the Federal Funds Rate and lagged Federal Funds Rate affect inflation dynamics.
- Federal Funds Rate: Research has shown that changes in the Federal Funds Rate, which is controlled by the Federal Reserve, can have a significant impact on inflation. A higher Federal Funds Rate tends to put downward pressure on inflation by making borrowing more expensive. Conversely, a lower rate can stimulate economic activity and potentially lead to higher inflation.
ARIMAX Model for 30-Year Treasury Yield: The Autoregressive Integrated Moving Average with Exogenous Variables model is an extension of the ARIMA model that incorporates external variables. When applied to the 30-Year Treasury Yield, it allows us to assess the influence of the Federal Funds Rate on long-term interest rates.
- Federal Funds Rate: Changes in the Federal Funds Rate can influence the 30-Year Treasury Yield. An increase in the Federal Funds Rate typically leads to higher long-term interest rates, including the 30-Year Treasury Yield. This relationship is essential for understanding the impact of monetary policy on the bond market.
VAR Model for Monetary Policy and Treasury Markets: Vector Autoregression (VAR) models are used to analyze the dynamic relationships among multiple time series variables. In the context of monetary policy and treasury markets, a VAR model can help us understand how different variables, including the Federal Funds Rate and various Treasury Yields, interact with each other.
- Fed Funds Rate, Treasury Yields: A VAR model that includes variables such as the Federal Funds Rate, 3-Month Yield, 6-Month Yield, 1-Year Yield, 5-Year Yield, 10-Year Yield, and 30-Year Yield can provide insights into the interconnectedness of these financial indicators. For example, an increase in the Federal Funds Rate can have ripple effects on short-term and long-term yields, impacting borrowing costs and investment decisions.
In summary, the literature review suggests that the Federal Funds Rate is a central variable that affects both inflation and interest rates. SARIMAX and ARIMAX models can be employed to capture these relationships, while a VAR model can provide a comprehensive view of the interactions among various monetary policy and treasury market variables. These modeling approaches are valuable tools for forecasting and policy analysis in the realm of monetary policy and treasury markets.
SARIMAX - Inflation vs. Fed Funds Rate
Data Processing
The plot shows the CPI and the Federal Funds Rate over time. The CPI has experienced a steady increase suggesting inflation growth, while the Federal Funds Rate has remained relatively stable, with only minor fluctuations over the same period.
Stationary Check
Augmented Dickey-Fuller Test Results for FedFundsRate :
Test Statistic: -8.408184
P-value: 0.01
Critical Values:
The time series FedFundsRate is stationary based on the ADF test.
Augmented Dickey-Fuller Test Results for L2FedFundsRate :
Test Statistic: -8.367854
P-value: 0.01
Critical Values:
The time series L2FedFundsRate is stationary based on the ADF test.
Augmented Dickey-Fuller Test Results for CPI :
Test Statistic: -5.128259
P-value: 0.01
Critical Values:
The time series CPI is stationary based on the ADF test.Model Fitting
auto.arima
auto.arima suggests a ARIMA model with seasonal components: SARIMAX(0,1,3)(0,0,2)[12]
Series: ts_combined[, "CPI"]
Regression with ARIMA(0,1,3)(0,0,2)[12] errors
Coefficients:
ma1 ma2 ma3 sma1 sma2 l0 l2
-0.4390 -0.3706 -0.0703 -0.1380 -0.0611 0.0279 0.0036
s.e. 0.0359 0.0350 0.0360 0.0361 0.0365 0.0291 0.0291
sigma^2 = 0.1436: log likelihood = -367.37
AIC=750.74 AICc=750.92 BIC=788.47
Training set error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 0.01415216 0.3770653 0.2166059 NaN Inf 0.6033404 -0.004461179Custom Model
The first regression model shows both the current and lagged Federal Funds Rate positively affecting the Consumer Price Index (CPI), with a slightly better fit than the second model. The second model, using only the current Federal Funds Rate, also indicates a positive effect on CPI but with a weaker fit. Both models confirm a significant relationship between the Federal Funds Rate and CPI.
1. CPI ~ FedFundsRate + L2FedFundsRate
Call:
lm(formula = CPI ~ FedFundsRate + L2FedFundsRate, data = ts_combined)
Residuals:
Min 1Q Median 3Q Max
-4.1340 -0.2589 -0.0540 0.1853 3.0811
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.33772 0.01681 20.085 <2e-16 ***
FedFundsRate 0.07781 0.03446 2.258 0.0242 *
L2FedFundsRate 0.05750 0.03444 1.670 0.0953 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4832 on 823 degrees of freedom
Multiple R-squared: 0.008938, Adjusted R-squared: 0.00653
F-statistic: 3.711 on 2 and 823 DF, p-value: 0.024862. CPI ~ FedFundsRate
Call:
lm(formula = CPI ~ FedFundsRate, data = ts_combined)
Residuals:
Min 1Q Median 3Q Max
-4.1371 -0.2586 -0.0499 0.1714 3.0894
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.33804 0.01683 20.08 <2e-16 ***
FedFundsRate 0.07401 0.03442 2.15 0.0318 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4837 on 824 degrees of freedom
Multiple R-squared: 0.005581, Adjusted R-squared: 0.004374
F-statistic: 4.624 on 1 and 824 DF, p-value: 0.03181ACF & PACF Plot
Significant lags shown in the first difference ACF and PACF plots: d=0,1, D=0,1, P=1,2, Q=1, p=1,2, q=0,1
Hyperparameter Optimization
Model fitting with minimum AIC:
1, 0, 2, 2, 1, 1, 782.400948943201, 815.306066710036, 782.540079377984
Model fitting with minimum AICc:
1, 0, 2, 2, 1, 1, 782.400948943201, 815.306066710036, 782.540079377984
Model fitting with minimum BIC:
1, 0, 2, 2, 1, 1, 782.400948943201, 815.306066710036, 782.540079377984Model Diagnostics
Both SARIMAX(1,0,2)(2,1,1)[12] and SARIMAX(0,1,3)(0,0,2)[12] evaluated through their diagnostic plots and statistical outputs suggest adequate fits for time series forecasting. The first model’s autocorrelations are within acceptable limits, and key coefficients are statistically significant, indicating a good fit, albeit with a non-significant constant term. The second model presents a marginally better fit, as evidenced by lower AIC and BIC values, and most coefficients are significant, though ma2 and sma2 are not. Both models have residuals that are approximately normally distributed and independent according to the Ljung-Box test, with the second model potentially being the preferred choice due to its lower complexity and better fit indicators.
[1] "Call:"
[2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "
[3] " xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
[4] " REPORT = 1, reltol = tol))"
[5] ""
[6] "Coefficients:"
[7] " ar1 ma1 ma2 sar1 sar2 sma1 constant"
[8] " 0.1529 -0.5980 -0.3007 -0.1782 -0.1219 -0.9520 0"
[9] "s.e. 0.0808 0.0773 0.0642 0.0396 0.0406 0.0246 0"
[10] ""
[11] "sigma^2 estimated as 0.1438: log likelihood = -383.89, aic = 783.78"
[12] ""
[13] "$degrees_of_freedom"
[14] "[1] 806"
[15] ""
[16] "$ttable"
[17] " Estimate SE t.value p.value"
[18] "ar1 0.1529 0.0808 1.8927 0.0588"
[19] "ma1 -0.5980 0.0773 -7.7396 0.0000"
[20] "ma2 -0.3007 0.0642 -4.6855 0.0000"
[21] "sar1 -0.1782 0.0396 -4.4990 0.0000"
[22] "sar2 -0.1219 0.0406 -3.0068 0.0027"
[23] "sma1 -0.9520 0.0246 -38.7447 0.0000"
[24] "constant 0.0000 0.0000 0.1604 0.8726"
[25] ""
[26] "$AIC"
[27] "[1] 0.9640587"
[28] ""
[29] "$AICc"
[30] "[1] 0.9642298"
[31] ""
[32] "$BIC"
[33] "[1] 1.010314"
[34] "" 
[1] "Call:"
[2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "
[3] " xreg = constant, transform.pars = trans, fixed = fixed, optim.control = list(trace = trc, "
[4] " REPORT = 1, reltol = tol))"
[5] ""
[6] "Coefficients:"
[7] " ma1 ma2 ma3 sma1 sma2 constant"
[8] " -1.4737 0.0731 0.4006 -0.1446 -0.0540 0"
[9] "s.e. 0.0318 0.0570 0.0313 0.0359 0.0364 0"
[10] ""
[11] "sigma^2 estimated as 0.1433: log likelihood = -375.27, aic = 764.54"
[12] ""
[13] "$degrees_of_freedom"
[14] "[1] 818"
[15] ""
[16] "$ttable"
[17] " Estimate SE t.value p.value"
[18] "ma1 -1.4737 0.0318 -46.2772 0.0000"
[19] "ma2 0.0731 0.0570 1.2818 0.2003"
[20] "ma3 0.4006 0.0313 12.7825 0.0000"
[21] "sma1 -0.1446 0.0359 -4.0336 0.0001"
[22] "sma2 -0.0540 0.0364 -1.4845 0.1381"
[23] "constant 0.0000 0.0000 0.1387 0.8897"
[24] ""
[25] "$AIC"
[26] "[1] 0.927841"
[27] ""
[28] "$AICc"
[29] "[1] 0.9279657"
[30] ""
[31] "$BIC"
[32] "[1] 0.9678885"
[33] "" Cross Validation
The SARIMAX(0,1,3)(0,0,2)[12] model demonstrates superior performance in cross-validation, exhibiting lower RMSE values during testing phases. This indicates a more accurate and reliable forecasting ability, particularly in capturing the seasonal trends and dependencies in the data.
Forecasting
The SARIMA model forecasts for the Federal Funds Rate and CPI suggest upcoming economic uncertainty. The Fed Funds Rate prediction shows potential for slight increases and a broad range of outcomes, signaling that interest rate policies may need to adapt to evolving economic conditions. The CPI forecast indicates rising volatility, with a trend that suggests inflation could be a concern, requiring careful monitoring and possible intervention from policymakers.
ARIMAX - 30-Year Treasury Yield vs. Fed Funds Rate
Data Processing
The plot shows the 30-Year Treasury Yield and the Federal Funds Rate over time. Both series appear to move together over the years, suggesting a relationship where changes in the Federal Funds Rate may be associated with similar movements in the long-term Treasury Yield.
Stationary Check
Augmented Dickey-Fuller Test Results for FedFundsRate :
Test Statistic: -6.327969
P-value: 0.01
Critical Values:
The time series FedFundsRate is stationary based on the ADF test.
Augmented Dickey-Fuller Test Results for treasury_yield_30 :
Test Statistic: -7.3842
P-value: 0.01
Critical Values:
The time series treasury_yield_30 is stationary based on the ADF test.Model Fitting
auto.arima
auto.arima suggests a ARIMA model: ARIMAX(0,0,2)
Series: ts_combined[, 2]
Regression with ARIMA(0,0,2) errors
Coefficients:
ma1 ma2 xreg
0.3316 -0.1327 0.1263
s.e. 0.0437 0.0429 0.0217
sigma^2 = 0.05685: log likelihood = 9.65
AIC=-11.29 AICc=-11.22 BIC=6.01
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set -0.005267137 0.2377827 0.1712959 -2.100979 260.2457 0.6323817
ACF1
Training set 0.001726696Custom Model
Call:
lm(formula = treasury_yield_30 ~ FedFundsRate, data = ts_combined)
Residuals:
Min 1Q Median 3Q Max
-1.09369 -0.13309 -0.00534 0.12427 1.49825
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.006401 0.010703 -0.598 0.55
FedFundsRate 0.149370 0.020198 7.395 5.23e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2528 on 556 degrees of freedom
Multiple R-squared: 0.08955, Adjusted R-squared: 0.08791
F-statistic: 54.69 on 1 and 556 DF, p-value: 5.23e-13ACF & PACF Plot
Significant lags shown in the first difference ACF and PACF plots: d=0,1, p=1,2, q=1,2
Hyperparameter Optimization
Model fitting with minimum AIC:
1, 0, 1, -6.77398623595946, 14.8478085759471, -6.66529058378554
Model fitting with minimum AICc:
1, 0, 1, -6.77398623595946, 14.8478085759471, -6.66529058378554
Model fitting with minimum BIC:
1, 0, 1, -6.77398623595946, 14.8478085759471, -6.66529058378554Model Diagnostics
The ARIMA model diagnostics for the 30-year Treasury yields and the Federal Funds Rate indicate effective fits with well-behaved residuals, suggesting that the models capture the essential patterns of these financial indicators. The significant terms in each model reflect the distinct behaviors of the yields and rates, with the 30-year Treasury likely exhibiting more long-term trends and the Federal Funds Rate responding to short-term influences. The slightly superior AIC and AICc values for the second model hint at a better fit for capturing the quick adjustments typical in the Federal Funds Rate.
[1] "Call:"
[2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "
[3] " xreg = xmean, include.mean = FALSE, transform.pars = trans, fixed = fixed, "
[4] " optim.control = list(trace = trc, REPORT = 1, reltol = tol))"
[5] ""
[6] "Coefficients:"
[7] " ar1 ma1 xmean"
[8] " -0.2930 0.6257 0.0001"
[9] "s.e. 0.0868 0.0686 0.0127"
[10] ""
[11] "sigma^2 estimated as 0.0568: log likelihood = 8.38, aic = -8.75"
[12] ""
[13] "$degrees_of_freedom"
[14] "[1] 555"
[15] ""
[16] "$ttable"
[17] " Estimate SE t.value p.value"
[18] "ar1 -0.2930 0.0868 -3.3746 0.0008"
[19] "ma1 0.6257 0.0686 9.1212 0.0000"
[20] "xmean 0.0001 0.0127 0.0112 0.9911"
[21] ""
[22] "$AIC"
[23] "[1] -0.01568965"
[24] ""
[25] "$AICc"
[26] "[1] -0.01561201"
[27] ""
[28] "$BIC"
[29] "[1] 0.01530934"
[30] "" 
[1] "Call:"
[2] "arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, Q), period = S), "
[3] " xreg = xmean, include.mean = FALSE, transform.pars = trans, fixed = fixed, "
[4] " optim.control = list(trace = trc, REPORT = 1, reltol = tol))"
[5] ""
[6] "Coefficients:"
[7] " ma1 ma2 xmean"
[8] " 0.3192 -0.1435 0.0001"
[9] "s.e. 0.0422 0.0418 0.0118"
[10] ""
[11] "sigma^2 estimated as 0.05663: log likelihood = 9.22, aic = -10.45"
[12] ""
[13] "$degrees_of_freedom"
[14] "[1] 555"
[15] ""
[16] "$ttable"
[17] " Estimate SE t.value p.value"
[18] "ma1 0.3192 0.0422 7.5680 0.0000"
[19] "ma2 -0.1435 0.0418 -3.4368 0.0006"
[20] "xmean 0.0001 0.0118 0.0073 0.9941"
[21] ""
[22] "$AIC"
[23] "[1] -0.01871965"
[24] ""
[25] "$AICc"
[26] "[1] -0.01864201"
[27] ""
[28] "$BIC"
[29] "[1] 0.01227934"
[30] "" Cross Validation
The ARIMAX(1,0,1) model demonstrates superior performance in cross-validation, exhibiting lower RMSE values during testing phases. This indicates a more accurate and reliable forecasting ability, particularly in capturing the seasonal trends and dependencies in the data.
Foreasting
The SARIMA model forecasts for the Federal Funds Rate and the 30-Year Treasury Yield depict stability in the near term, with an increasing range of outcomes as we move further into the future. The Fed Funds Rate is shown to have a stable past with minor fluctuations, but the forecast suggests uncertainty, implying that rates could either rise or fall. Similarly, the Treasury Yield forecast indicates relatively stable past rates but projects increasing uncertainty. This suggests that the long-term lending market could face varied conditions, potentially affecting investment, inflation, and economic growth expectations.
VAR - Monetary Policy and Treasury Markets
Variables: Fed Funds Rate, 3-Month Yield, 6-Month Yield, 1-Year Yield, 5-Year Yield, 10-Year Yield, 30-Year Yield.
Data Processing
This graph illustrates the Federal Funds Rate (FFR) and various Treasury yields (ranging from 3 months to 30 years) over time. The general trend shows that Treasury yields across different maturities tend to move in the same direction as the Federal Funds Rate, with the highest yields typically associated with longer maturities.
The pairplot indicates a generally positive correlation among all interest rate variables, from the Fed Funds Rate to the 30-Year Yield—when one goes up, others often do too. The time series plot shows how these rates have moved over time, sometimes in sync, reflecting similar economic forces, and sometimes diverging due to different market expectations for the short and long term.
Model Fitting
VAR Selection
VAR Model Lag Value Selection:AIC(n) HQ(n) SC(n) FPE(n)
2 2 2 2 VAR(1)
The VAR(1) model analysis underscores the significant economic impact of the Federal Funds Rate (FFR) on various Treasury yields. It shows how changes in the FFR ripple through both short-term and long-term interest rates, influencing the entire yield curve. Short-term rates respond quickly to monetary policy shifts, while long-term rates reflect market expectations about future economic trends. This interconnection highlights the far-reaching effects of central bank policies on different aspects of the economy, from consumer borrowing to long-term investments.
VAR Estimation Results:
=========================
Endogenous variables: FFR, T3Mon, T6Mon, T1Yr, T5Yr, T10Yr, T30Yr
Deterministic variables: both
Sample size: 499
Log Likelihood: 3050.254
Roots of the characteristic polynomial:
0.9735 0.9544 0.8926 0.8926 0.7743 0.7743 0.6561
Call:
vars::VAR(y = ts_combined, p = 1, type = c("both"))
Estimation results for equation FFR:
====================================
FFR = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.7746280 0.0844082 9.177 < 2e-16 ***
T3Mon.l1 0.3511073 0.1123396 3.125 0.00188 **
T6Mon.l1 -0.4857568 0.2165015 -2.244 0.02530 *
T1Yr.l1 0.7110530 0.1472259 4.830 1.83e-06 ***
T5Yr.l1 -0.4046803 0.1083445 -3.735 0.00021 ***
T10Yr.l1 0.3498053 0.1625501 2.152 0.03189 *
T30Yr.l1 -0.1491100 0.0916170 -1.628 0.10427
const 0.3653373 0.1379056 2.649 0.00833 **
trend -0.0005838 0.0002444 -2.389 0.01728 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2337 on 490 degrees of freedom
Multiple R-Squared: 0.995, Adjusted R-squared: 0.9949
F-statistic: 1.215e+04 on 8 and 490 DF, p-value: < 2.2e-16
Estimation results for equation T3Mon:
======================================
T3Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0685659 0.0639188 1.073 0.2839
T3Mon.l1 0.9409624 0.0850701 11.061 <2e-16 ***
T6Mon.l1 -0.3813387 0.1639475 -2.326 0.0204 *
T1Yr.l1 0.1451754 0.1114880 1.302 0.1935
T5Yr.l1 0.1128966 0.0820447 1.376 0.1694
T10Yr.l1 -0.0870301 0.1230924 -0.707 0.4799
T30Yr.l1 0.0067650 0.0693777 0.098 0.9224
const -0.0440213 0.1044301 -0.422 0.6735
trend 0.0002047 0.0001851 1.106 0.2692
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.177 on 490 degrees of freedom
Multiple R-Squared: 0.7574, Adjusted R-squared: 0.7534
F-statistic: 191.2 on 8 and 490 DF, p-value: < 2.2e-16
Estimation results for equation T6Mon:
======================================
T6Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0641697 0.0668795 0.959 0.337788
T3Mon.l1 0.3186130 0.0890105 3.579 0.000379 ***
T6Mon.l1 0.1069623 0.1715415 0.624 0.533223
T1Yr.l1 0.3236682 0.1166521 2.775 0.005737 **
T5Yr.l1 0.0984928 0.0858450 1.147 0.251804
T10Yr.l1 -0.0589513 0.1287940 -0.458 0.647358
T30Yr.l1 -0.0020282 0.0725913 -0.028 0.977721
const -0.1137577 0.1092673 -1.041 0.298345
trend 0.0003620 0.0001936 1.869 0.062189 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1852 on 490 degrees of freedom
Multiple R-Squared: 0.7376, Adjusted R-squared: 0.7333
F-statistic: 172.2 on 8 and 490 DF, p-value: < 2.2e-16
Estimation results for equation T1Yr:
=====================================
T1Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0826055 0.0763778 1.082 0.279990
T3Mon.l1 0.3137411 0.1016520 3.086 0.002140 **
T6Mon.l1 -0.6752194 0.1959042 -3.447 0.000616 ***
T1Yr.l1 1.0984511 0.1332193 8.245 1.52e-15 ***
T5Yr.l1 0.1476077 0.0980369 1.506 0.132806
T10Yr.l1 -0.0847203 0.1470856 -0.576 0.564884
T30Yr.l1 0.0067212 0.0829008 0.081 0.935415
const -0.0983891 0.1247857 -0.788 0.430806
trend 0.0003020 0.0002211 1.366 0.172658
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2115 on 490 degrees of freedom
Multiple R-Squared: 0.7854, Adjusted R-squared: 0.7819
F-statistic: 224.1 on 8 and 490 DF, p-value: < 2.2e-16
Estimation results for equation T5Yr:
=====================================
T5Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0690664 0.1036273 0.666 0.50541
T3Mon.l1 0.3976029 0.1379186 2.883 0.00411 **
T6Mon.l1 -0.7755880 0.2657974 -2.918 0.00369 **
T1Yr.l1 0.0803278 0.1807482 0.444 0.65694
T5Yr.l1 1.1045613 0.1330138 8.304 9.89e-16 ***
T10Yr.l1 -0.1243113 0.1995617 -0.623 0.53363
T30Yr.l1 0.0707768 0.1124776 0.629 0.52948
const -0.1467030 0.1693058 -0.866 0.38664
trend 0.0002964 0.0003000 0.988 0.32372
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.287 on 490 degrees of freedom
Multiple R-Squared: 0.9163, Adjusted R-squared: 0.9149
F-statistic: 670.3 on 8 and 490 DF, p-value: < 2.2e-16
Estimation results for equation T10Yr:
======================================
T10Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 -0.0293720 0.0894883 -0.328 0.743
T3Mon.l1 0.7075690 0.1191008 5.941 5.38e-09 ***
T6Mon.l1 -1.1130431 0.2295317 -4.849 1.67e-06 ***
T1Yr.l1 0.6360693 0.1560868 4.075 5.36e-05 ***
T5Yr.l1 -0.1352939 0.1148652 -1.178 0.239
T10Yr.l1 1.0141433 0.1723333 5.885 7.40e-09 ***
T30Yr.l1 0.0074114 0.0971310 0.076 0.939
const 0.2207933 0.1462055 1.510 0.132
trend -0.0003204 0.0002591 -1.237 0.217
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2478 on 490 degrees of freedom
Multiple R-Squared: 0.9934, Adjusted R-squared: 0.9933
F-statistic: 9219 on 8 and 490 DF, p-value: < 2.2e-16
Estimation results for equation T30Yr:
======================================
T30Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 -0.0927739 0.0796491 -1.165 0.244674
T3Mon.l1 0.6233762 0.1060058 5.881 7.57e-09 ***
T6Mon.l1 -0.9272946 0.2042948 -4.539 7.12e-06 ***
T1Yr.l1 0.5013582 0.1389251 3.609 0.000339 ***
T5Yr.l1 -0.1750598 0.1022359 -1.712 0.087472 .
T10Yr.l1 0.1989505 0.1533853 1.297 0.195220
T30Yr.l1 0.8773440 0.0864515 10.148 < 2e-16 ***
const 0.2855076 0.1301303 2.194 0.028703 *
trend -0.0003541 0.0002306 -1.536 0.125266
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2206 on 490 degrees of freedom
Multiple R-Squared: 0.9938, Adjusted R-squared: 0.9937
F-statistic: 9842 on 8 and 490 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
FFR T3Mon T6Mon T1Yr T5Yr T10Yr T30Yr
FFR 0.054637 -0.006841 -0.01050 -0.01507 -0.03161 0.01695 0.01364
T3Mon -0.006841 0.031331 0.02878 0.02718 0.02575 0.01369 0.01077
T6Mon -0.010501 0.028779 0.03430 0.03661 0.03848 0.02227 0.01700
T1Yr -0.015067 0.027176 0.03661 0.04474 0.05183 0.03040 0.02336
T5Yr -0.031615 0.025752 0.03848 0.05183 0.08235 0.04693 0.03773
T10Yr 0.016950 0.013694 0.02227 0.03040 0.04693 0.06141 0.05193
T30Yr 0.013640 0.010769 0.01700 0.02336 0.03773 0.05193 0.04865
Correlation matrix of residuals:
FFR T3Mon T6Mon T1Yr T5Yr T10Yr T30Yr
FFR 1.0000 -0.1653 -0.2426 -0.3048 -0.4713 0.2926 0.2646
T3Mon -0.1653 1.0000 0.8779 0.7259 0.5070 0.3122 0.2758
T6Mon -0.2426 0.8779 1.0000 0.9345 0.7239 0.4853 0.4162
T1Yr -0.3048 0.7259 0.9345 1.0000 0.8540 0.5800 0.5008
T5Yr -0.4713 0.5070 0.7239 0.8540 1.0000 0.6600 0.5962
T10Yr 0.2926 0.3122 0.4853 0.5800 0.6600 1.0000 0.9500
T30Yr 0.2646 0.2758 0.4162 0.5008 0.5962 0.9500 1.0000VAR(2)
The VAR(2) model’s analysis shows how changes in the FFR, a key monetary policy tool, not only have immediate impacts but also exert influence over longer periods through their interactions with short-term and long-term yields. These interdependencies between different segments of the yield curve are crucial for understanding market expectations and economic conditions.
VAR Estimation Results:
=========================
Endogenous variables: FFR, T3Mon, T6Mon, T1Yr, T5Yr, T10Yr, T30Yr
Deterministic variables: both
Sample size: 498
Log Likelihood: 3235.419
Roots of the characteristic polynomial:
0.9678 0.9101 0.8833 0.8592 0.8073 0.8073 0.6482 0.3795 0.3541 0.2308 0.1858 0.1302 0.1302 0.01743
Call:
vars::VAR(y = ts_combined, p = 2, type = "both")
Estimation results for equation FFR:
====================================
FFR = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.9521757 0.1352793 7.039 6.71e-12 ***
T3Mon.l1 0.3785554 0.1352109 2.800 0.005320 **
T6Mon.l1 -0.6044160 0.2550213 -2.370 0.018178 *
T1Yr.l1 0.6883991 0.2052075 3.355 0.000857 ***
T5Yr.l1 -0.5577872 0.1724705 -3.234 0.001304 **
T10Yr.l1 0.7228538 0.2362035 3.060 0.002334 **
T30Yr.l1 -0.2982019 0.1544073 -1.931 0.054036 .
FFR.l2 -0.0556846 0.1341871 -0.415 0.678343
T3Mon.l2 -0.1512887 0.1367911 -1.106 0.269285
T6Mon.l2 0.2900315 0.2558634 1.134 0.257551
T1Yr.l2 -0.2381943 0.2076678 -1.147 0.251952
T5Yr.l2 0.3657711 0.1735552 2.108 0.035588 *
T10Yr.l2 -0.5464220 0.2367433 -2.308 0.021417 *
T30Yr.l2 0.1849556 0.1528606 1.210 0.226886
const 0.4594991 0.1243409 3.695 0.000245 ***
trend -0.0007465 0.0002202 -3.390 0.000755 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2055 on 482 degrees of freedom
Multiple R-Squared: 0.9961, Adjusted R-squared: 0.996
F-statistic: 8211 on 15 and 482 DF, p-value: < 2.2e-16
Estimation results for equation T3Mon:
======================================
T3Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 2.233e-01 1.154e-01 1.936 0.0535 .
T3Mon.l1 1.106e+00 1.153e-01 9.590 <2e-16 ***
T6Mon.l1 -5.292e-01 2.175e-01 -2.433 0.0153 *
T1Yr.l1 -6.771e-03 1.750e-01 -0.039 0.9692
T5Yr.l1 2.921e-01 1.471e-01 1.986 0.0476 *
T10Yr.l1 -1.645e-01 2.014e-01 -0.817 0.4144
T30Yr.l1 2.998e-02 1.317e-01 0.228 0.8200
FFR.l2 -1.560e-01 1.144e-01 -1.363 0.1735
T3Mon.l2 -2.750e-01 1.167e-01 -2.358 0.0188 *
T6Mon.l2 3.412e-01 2.182e-01 1.564 0.1185
T1Yr.l2 5.042e-02 1.771e-01 0.285 0.7760
T5Yr.l2 -1.663e-01 1.480e-01 -1.124 0.2616
T10Yr.l2 7.890e-02 2.019e-01 0.391 0.6961
T30Yr.l2 -3.163e-02 1.304e-01 -0.243 0.8084
const 1.947e-02 1.060e-01 0.184 0.8544
trend 7.458e-05 1.878e-04 0.397 0.6914
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1752 on 482 degrees of freedom
Multiple R-Squared: 0.7659, Adjusted R-squared: 0.7586
F-statistic: 105.1 on 15 and 482 DF, p-value: < 2.2e-16
Estimation results for equation T6Mon:
======================================
T6Mon = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.2086962 0.1211763 1.722 0.08567 .
T3Mon.l1 0.3987475 0.1211149 3.292 0.00107 **
T6Mon.l1 -0.0179431 0.2284349 -0.079 0.93742
T1Yr.l1 0.2984660 0.1838143 1.624 0.10509
T5Yr.l1 0.2974574 0.1544902 1.925 0.05477 .
T10Yr.l1 -0.2257951 0.2115789 -1.067 0.28642
T30Yr.l1 0.1240212 0.1383101 0.897 0.37033
FFR.l2 -0.1523969 0.1201978 -1.268 0.20545
T3Mon.l2 -0.2123357 0.1225304 -1.733 0.08375 .
T6Mon.l2 0.3498190 0.2291892 1.526 0.12758
T1Yr.l2 -0.0819299 0.1860181 -0.440 0.65982
T5Yr.l2 -0.2028815 0.1554618 -1.305 0.19251
T10Yr.l2 0.1678978 0.2120625 0.792 0.42890
T30Yr.l2 -0.1233282 0.1369246 -0.901 0.36820
const -0.0849029 0.1113782 -0.762 0.44626
trend 0.0002823 0.0001972 1.431 0.15304
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1841 on 482 degrees of freedom
Multiple R-Squared: 0.745, Adjusted R-squared: 0.7371
F-statistic: 93.9 on 15 and 482 DF, p-value: < 2.2e-16
Estimation results for equation T1Yr:
=====================================
T1Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.2113325 0.1351125 1.564 0.1184
T3Mon.l1 0.2431981 0.1350441 1.801 0.0723 .
T6Mon.l1 -0.6036530 0.2547069 -2.370 0.0182 *
T1Yr.l1 1.0483188 0.2049545 5.115 4.54e-07 ***
T5Yr.l1 0.3915911 0.1722578 2.273 0.0234 *
T10Yr.l1 -0.3111313 0.2359123 -1.319 0.1878
T30Yr.l1 0.2146394 0.1542169 1.392 0.1646
FFR.l2 -0.1460338 0.1340216 -1.090 0.2764
T3Mon.l2 -0.0598441 0.1366224 -0.438 0.6616
T6Mon.l2 0.1270787 0.2555479 0.497 0.6192
T1Yr.l2 -0.0178044 0.2074118 -0.086 0.9316
T5Yr.l2 -0.2756804 0.1733412 -1.590 0.1124
T10Yr.l2 0.2243337 0.2364514 0.949 0.3432
T30Yr.l2 -0.1870179 0.1526721 -1.225 0.2212
const -0.1127772 0.1241876 -0.908 0.3643
trend 0.0002820 0.0002199 1.282 0.2003
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2052 on 482 degrees of freedom
Multiple R-Squared: 0.8012, Adjusted R-squared: 0.795
F-statistic: 129.5 on 15 and 482 DF, p-value: < 2.2e-16
Estimation results for equation T5Yr:
=====================================
T5Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.1372860 0.1775727 0.773 0.4398
T3Mon.l1 0.2149833 0.1774828 1.211 0.2264
T6Mon.l1 -0.7292255 0.3347505 -2.178 0.0299 *
T1Yr.l1 0.0805036 0.2693630 0.299 0.7652
T5Yr.l1 1.4981225 0.2263912 6.617 9.75e-11 ***
T10Yr.l1 -0.5528881 0.3100496 -1.783 0.0752 .
T30Yr.l1 0.4731419 0.2026808 2.334 0.0200 *
FFR.l2 -0.1151528 0.1761389 -0.654 0.5136
T3Mon.l2 0.0840939 0.1795571 0.468 0.6398
T6Mon.l2 0.1514553 0.3358558 0.451 0.6522
T1Yr.l2 -0.0106697 0.2725925 -0.039 0.9688
T5Yr.l2 -0.4903611 0.2278150 -2.152 0.0319 *
T10Yr.l2 0.4558447 0.3107581 1.467 0.1431
T30Yr.l2 -0.3773127 0.2006505 -1.880 0.0606 .
const -0.1574989 0.1632145 -0.965 0.3350
trend 0.0002555 0.0002890 0.884 0.3771
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2697 on 482 degrees of freedom
Multiple R-Squared: 0.927, Adjusted R-squared: 0.9248
F-statistic: 408.2 on 15 and 482 DF, p-value: < 2.2e-16
Estimation results for equation T10Yr:
======================================
T10Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.1913693 0.1548277 1.236 0.217055
T3Mon.l1 0.5707830 0.1547494 3.688 0.000251 ***
T6Mon.l1 -1.0672439 0.2918729 -3.657 0.000284 ***
T1Yr.l1 0.4950770 0.2348608 2.108 0.035550 *
T5Yr.l1 0.1751568 0.1973932 0.887 0.375333
T10Yr.l1 0.7822002 0.2703359 2.893 0.003983 **
T30Yr.l1 0.3752108 0.1767197 2.123 0.034246 *
FFR.l2 -0.1847856 0.1535776 -1.203 0.229487
T3Mon.l2 -0.0432282 0.1565579 -0.276 0.782577
T6Mon.l2 0.2244892 0.2928366 0.767 0.443693
T1Yr.l2 -0.0270054 0.2376766 -0.114 0.909584
T5Yr.l2 -0.2569851 0.1986346 -1.294 0.196369
T10Yr.l2 0.1525009 0.2709536 0.563 0.573812
T30Yr.l2 -0.3338779 0.1749495 -1.908 0.056930 .
const 0.3023725 0.1423086 2.125 0.034114 *
trend -0.0005014 0.0002520 -1.990 0.047197 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2352 on 482 degrees of freedom
Multiple R-Squared: 0.994, Adjusted R-squared: 0.9939
F-statistic: 5364 on 15 and 482 DF, p-value: < 2.2e-16
Estimation results for equation T30Yr:
======================================
T30Yr = FFR.l1 + T3Mon.l1 + T6Mon.l1 + T1Yr.l1 + T5Yr.l1 + T10Yr.l1 + T30Yr.l1 + FFR.l2 + T3Mon.l2 + T6Mon.l2 + T1Yr.l2 + T5Yr.l2 + T10Yr.l2 + T30Yr.l2 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0248940 0.1385365 0.180 0.857470
T3Mon.l1 0.4661740 0.1384664 3.367 0.000822 ***
T6Mon.l1 -0.8266782 0.2611616 -3.165 0.001647 **
T1Yr.l1 0.3821162 0.2101484 1.818 0.069636 .
T5Yr.l1 -0.0020661 0.1766232 -0.012 0.990672
T10Yr.l1 -0.0368552 0.2418907 -0.152 0.878964
T30Yr.l1 1.3033321 0.1581250 8.242 1.61e-15 ***
FFR.l2 -0.0848315 0.1374179 -0.617 0.537312
T3Mon.l2 0.0420729 0.1400847 0.300 0.764048
T6Mon.l2 0.0559831 0.2620239 0.214 0.830905
T1Yr.l2 0.0093640 0.2126679 0.044 0.964898
T5Yr.l2 -0.1230581 0.1777340 -0.692 0.489037
T10Yr.l2 0.1789105 0.2424435 0.738 0.460906
T30Yr.l2 -0.4114694 0.1565411 -2.629 0.008850 **
const 0.3683984 0.1273347 2.893 0.003987 **
trend -0.0005069 0.0002255 -2.248 0.025031 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2104 on 482 degrees of freedom
Multiple R-Squared: 0.9944, Adjusted R-squared: 0.9942
F-statistic: 5664 on 15 and 482 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
FFR T3Mon T6Mon T1Yr T5Yr T10Yr T30Yr
FFR 0.042220 -0.009279 -0.01068 -0.01265 -0.02491 0.01365 0.01039
T3Mon -0.009279 0.030703 0.02875 0.02781 0.02727 0.01307 0.01034
T6Mon -0.010676 0.028749 0.03388 0.03573 0.03701 0.02067 0.01588
T1Yr -0.012650 0.027810 0.03573 0.04212 0.04704 0.02775 0.02153
T5Yr -0.024911 0.027270 0.03701 0.04704 0.07275 0.04316 0.03511
T10Yr 0.013648 0.013072 0.02067 0.02775 0.04316 0.05530 0.04687
T30Yr 0.010394 0.010343 0.01588 0.02153 0.03511 0.04687 0.04428
Correlation matrix of residuals:
FFR T3Mon T6Mon T1Yr T5Yr T10Yr T30Yr
FFR 1.0000 -0.2577 -0.2823 -0.3000 -0.4495 0.2824 0.2404
T3Mon -0.2577 1.0000 0.8914 0.7734 0.5770 0.3172 0.2805
T6Mon -0.2823 0.8914 1.0000 0.9459 0.7455 0.4775 0.4100
T1Yr -0.3000 0.7734 0.9459 1.0000 0.8499 0.5750 0.4987
T5Yr -0.4495 0.5770 0.7455 0.8499 1.0000 0.6805 0.6187
T10Yr 0.2824 0.3172 0.4775 0.5750 0.6805 1.0000 0.9472
T30Yr 0.2404 0.2805 0.4100 0.4987 0.6187 0.9472 1.0000Cross Validation
The plots compare the cross-validation RMSE for VAR(1) and VAR(2) models across different financial variables over time. The consistently lower RMSE values for VAR(1) indicate that it provides a better fit or more accurate forecasts for these series.
Forecasting
The forecast plots generated from the VAR(1) model for various interest rates, including the Fed Funds Rate and various Treasury yields, present a detailed picture of expected movements and uncertainties within the financial market. These plots suggest that each interest rate may follow a different path in the future, yet they are all interrelated. Typically, the Fed Funds Rate, which is set by the Federal Reserve, influences the other yields, as it reflects the cost of borrowing funds overnight. Treasury yields of different maturities react to this benchmark rate but also incorporate market expectations for inflation and economic growth over various horizons. The plots show that while forecasts are made for each rate individually, the market’s future expectations for each can diverge due to different influencing factors, including policy decisions, economic outlook, and investor sentiment.
VAR - Monetary Policy and Macroeconomic Indicators
Variables: FFR, GDP, Unemployment Rate, CPI, 30-Year Mortgage Rate
Data Processing
The pair plot illustrates the disparity in scales among different economic indicators, making it difficult to discern relationships directly due to the dominance of variables like GDP. The time series plot reveals distinct trends over time, such as the steady increase in GDP and the cyclical nature of the unemployment rate, alongside the long-term rise in CPI and the decline in the mortgage rate.
Model Fitting
The progression from VAR(1) through VAR(3), VAR(7), to VAR(10) models shows an evolving understanding of the interrelationships among the key economic indicators: Federal Funds Rate, Gross Domestic Product, Unemployment Rate, Consumer Price Index, and 30-Year Mortgage Rate.
In all models, the persistent influence of FFR on MORT is evident, highlighting the impact of monetary policy on long-term interest rates. The GDP equations across these models consistently show its own past values as significant predictors, indicating economic growth’s self-sustaining nature.
As the lag order increases, the models reveal more complex dynamics. For example, the VAR(10) model captures longer-term influences, showing how variables like UMP and CPI impact others over extended periods. However, with higher-order VAR models, the risk of overfitting increases, and some coefficients may become statistically insignificant, suggesting that not all lagged values meaningfully contribute to the model.
The overall fit of the model improves with higher lag orders, as seen in the increasing R-squared values, but this comes with the trade-off of increased complexity and potential redundancy in lagged terms. The choice of the model should balance the complexity with interpretability and the specific requirements of the analysis.
VAR Selection
VAR Model Lag Value Selection:AIC(n) HQ(n) SC(n) FPE(n)
10 7 3 10 VAR(1)
VAR Estimation Results:
=========================
Endogenous variables: FFR, GDP, UMP, CPI, MORT
Deterministic variables: both
Sample size: 624
Log Likelihood: -4567.829
Roots of the characteristic polynomial:
1 1 0.9706 0.9706 0.9289
Call:
vars::VAR(y = ts_combined, p = 1, type = c("both"))
Estimation results for equation FFR:
====================================
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 9.779e-01 1.676e-02 58.357 <2e-16 ***
GDP.l1 5.989e-05 5.117e-05 1.170 0.242
UMP.l1 -2.315e-02 1.615e-02 -1.433 0.152
CPI.l1 4.340e-03 5.346e-03 0.812 0.417
MORT.l1 -5.592e-03 2.425e-02 -0.231 0.818
const 3.984e-02 3.771e-01 0.106 0.916
trend -3.820e-03 2.524e-03 -1.514 0.131
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5307 on 617 degrees of freedom
Multiple R-Squared: 0.9823, Adjusted R-squared: 0.9821
F-statistic: 5711 on 6 and 617 DF, p-value: < 2.2e-16
Estimation results for equation GDP:
====================================
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 1.360650 1.805933 0.753 0.451
GDP.l1 1.000391 0.005515 181.407 < 2e-16 ***
UMP.l1 9.904724 1.740747 5.690 1.96e-08 ***
CPI.l1 -0.793751 0.576166 -1.378 0.169
MORT.l1 -3.406546 2.612830 -1.304 0.193
const 3.647914 40.636318 0.090 0.928
trend 0.326185 0.271960 1.199 0.231
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 57.19 on 617 degrees of freedom
Multiple R-Squared: 0.9999, Adjusted R-squared: 0.9999
F-statistic: 7.81e+05 on 6 and 617 DF, p-value: < 2.2e-16
Estimation results for equation UMP:
====================================
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 -4.876e-03 1.470e-02 -0.332 0.7402
GDP.l1 -1.039e-04 4.488e-05 -2.315 0.0209 *
UMP.l1 9.392e-01 1.417e-02 66.291 <2e-16 ***
CPI.l1 -2.081e-03 4.689e-03 -0.444 0.6573
MORT.l1 1.376e-02 2.127e-02 0.647 0.5177
const 8.324e-01 3.307e-01 2.517 0.0121 *
trend 3.558e-03 2.213e-03 1.608 0.1085
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4655 on 617 degrees of freedom
Multiple R-Squared: 0.9261, Adjusted R-squared: 0.9254
F-statistic: 1289 on 6 and 617 DF, p-value: < 2.2e-16
Estimation results for equation CPI:
====================================
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 5.849e-02 1.559e-02 3.753 0.000191 ***
GDP.l1 2.482e-04 4.759e-05 5.215 2.51e-07 ***
UMP.l1 3.528e-02 1.502e-02 2.348 0.019168 *
CPI.l1 1.016e+00 4.972e-03 204.346 < 2e-16 ***
MORT.l1 -2.937e-02 2.255e-02 -1.303 0.193200
const -1.834e+00 3.507e-01 -5.229 2.34e-07 ***
trend -1.194e-02 2.347e-03 -5.085 4.87e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4936 on 617 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 0.9999
F-statistic: 2.073e+06 on 6 and 617 DF, p-value: < 2.2e-16
Estimation results for equation MORT:
=====================================
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 4.510e-02 8.448e-03 5.339 1.32e-07 ***
GDP.l1 1.213e-05 2.580e-05 0.470 0.6385
UMP.l1 5.126e-03 8.143e-03 0.629 0.5293
CPI.l1 5.699e-03 2.695e-03 2.114 0.0349 *
MORT.l1 9.366e-01 1.222e-02 76.624 < 2e-16 ***
const 2.334e-02 1.901e-01 0.123 0.9023
trend -2.683e-03 1.272e-03 -2.109 0.0354 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2675 on 617 degrees of freedom
Multiple R-Squared: 0.9934, Adjusted R-squared: 0.9933
F-statistic: 1.543e+04 on 6 and 617 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
FFR GDP UMP CPI MORT
FFR 0.28161 3.9457 -0.031377 0.02583 0.068411
GDP 3.94574 3270.6260 -14.698226 5.69067 0.482888
UMP -0.03138 -14.6982 0.216650 -0.05230 -0.007939
CPI 0.02583 5.6907 -0.052298 0.24360 0.022234
MORT 0.06841 0.4829 -0.007939 0.02223 0.071575
Correlation matrix of residuals:
FFR GDP UMP CPI MORT
FFR 1.00000 0.13001 -0.12703 0.09863 0.48185
GDP 0.13001 1.00000 -0.55217 0.20161 0.03156
UMP -0.12703 -0.55217 1.00000 -0.22765 -0.06376
CPI 0.09863 0.20161 -0.22765 1.00000 0.16838
MORT 0.48185 0.03156 -0.06376 0.16838 1.00000VAR(3)
VAR Estimation Results:
=========================
Endogenous variables: FFR, GDP, UMP, CPI, MORT
Deterministic variables: both
Sample size: 622
Log Likelihood: -3937.128
Roots of the characteristic polynomial:
0.9999 0.9999 0.9794 0.9794 0.8771 0.6467 0.6467 0.5721 0.5721 0.4166 0.4166 0.3605 0.3605 0.2805 0.2805
Call:
vars::VAR(y = ts_combined, p = 3, type = "both")
Estimation results for equation FFR:
====================================
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 1.4252924 0.0447353 31.861 < 2e-16 ***
GDP.l1 0.0007745 0.0005562 1.393 0.16426
UMP.l1 -0.0143692 0.0517842 -0.277 0.78151
CPI.l1 0.0255165 0.0454245 0.562 0.57450
MORT.l1 0.0401763 0.0886834 0.453 0.65069
FFR.l2 -0.5852555 0.0715553 -8.179 1.69e-15 ***
GDP.l2 -0.0007781 0.0011277 -0.690 0.49044
UMP.l2 -0.0100167 0.0772807 -0.130 0.89691
CPI.l2 0.0004464 0.0776948 0.006 0.99542
MORT.l2 -0.2413403 0.1387852 -1.739 0.08255 .
FFR.l3 0.1087544 0.0455998 2.385 0.01739 *
GDP.l3 0.0000650 0.0006748 0.096 0.92329
UMP.l3 -0.0077867 0.0667742 -0.117 0.90721
CPI.l3 -0.0256091 0.0468321 -0.547 0.58470
MORT.l3 0.2412074 0.0855309 2.820 0.00496 **
const -0.0424590 0.3558114 -0.119 0.90505
trend -0.0022319 0.0024551 -0.909 0.36368
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4738 on 605 degrees of freedom
Multiple R-Squared: 0.9862, Adjusted R-squared: 0.9858
F-statistic: 2697 on 16 and 605 DF, p-value: < 2.2e-16
Estimation results for equation GDP:
====================================
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 6.78267 3.28454 2.065 0.0393 *
GDP.l1 1.84556 0.04084 45.195 < 2e-16 ***
UMP.l1 79.73754 3.80208 20.972 < 2e-16 ***
CPI.l1 -2.80974 3.33514 -0.842 0.3999
MORT.l1 5.86310 6.51128 0.900 0.3682
FFR.l2 -9.18057 5.25371 -1.747 0.0811 .
GDP.l2 -0.71591 0.08279 -8.647 < 2e-16 ***
UMP.l2 -54.51339 5.67408 -9.607 < 2e-16 ***
CPI.l2 3.82624 5.70448 0.671 0.5026
MORT.l2 -4.91168 10.18984 -0.482 0.6300
FFR.l3 1.31237 3.34801 0.392 0.6952
GDP.l3 -0.13406 0.04954 -2.706 0.0070 **
UMP.l3 -23.55930 4.90267 -4.805 1.95e-06 ***
CPI.l3 -1.01128 3.43849 -0.294 0.7688
MORT.l3 -1.38550 6.27982 -0.221 0.8255
const 22.81706 26.12426 0.873 0.3828
trend 0.10449 0.18026 0.580 0.5624
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 34.79 on 605 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 7.861e+05 on 16 and 605 DF, p-value: < 2.2e-16
Estimation results for equation UMP:
====================================
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 -0.0843618 0.0331153 -2.548 0.01110 *
GDP.l1 -0.0025889 0.0004117 -6.288 6.16e-10 ***
UMP.l1 0.5057201 0.0383332 13.193 < 2e-16 ***
CPI.l1 -0.0438201 0.0336255 -1.303 0.19301
MORT.l1 0.0312865 0.0656479 0.477 0.63383
FFR.l2 0.1236302 0.0529688 2.334 0.01992 *
GDP.l2 -0.0026738 0.0008347 -3.203 0.00143 **
UMP.l2 0.0604795 0.0572070 1.057 0.29084
CPI.l2 0.0202873 0.0575136 0.353 0.72441
MORT.l2 -0.1050433 0.1027357 -1.022 0.30697
FFR.l3 -0.0289493 0.0337553 -0.858 0.39144
GDP.l3 0.0052247 0.0004995 10.460 < 2e-16 ***
UMP.l3 0.4191785 0.0494296 8.480 < 2e-16 ***
CPI.l3 0.0176832 0.0346675 0.510 0.61018
MORT.l3 0.0738715 0.0633142 1.167 0.24377
const 0.5413245 0.2633894 2.055 0.04029 *
trend 0.0036114 0.0018174 1.987 0.04736 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3507 on 605 degrees of freedom
Multiple R-Squared: 0.9589, Adjusted R-squared: 0.9578
F-statistic: 881.3 on 16 and 605 DF, p-value: < 2.2e-16
Estimation results for equation CPI:
====================================
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0590672 0.0400603 1.474 0.140878
GDP.l1 0.0016346 0.0004981 3.282 0.001090 **
UMP.l1 0.1635305 0.0463725 3.526 0.000453 ***
CPI.l1 1.5014443 0.0406774 36.911 < 2e-16 ***
MORT.l1 0.0891795 0.0794156 1.123 0.261905
FFR.l2 -0.0502386 0.0640775 -0.784 0.433330
GDP.l2 -0.0006613 0.0010098 -0.655 0.512788
UMP.l2 -0.0721825 0.0692046 -1.043 0.297350
CPI.l2 -0.5992755 0.0695754 -8.613 < 2e-16 ***
MORT.l2 -0.0699906 0.1242816 -0.563 0.573534
FFR.l3 0.0245324 0.0408344 0.601 0.548213
GDP.l3 -0.0008465 0.0006043 -1.401 0.161739
UMP.l3 -0.0773094 0.0597960 -1.293 0.196544
CPI.l3 0.1054326 0.0419380 2.514 0.012195 *
MORT.l3 -0.0343605 0.0765926 -0.449 0.653871
const -0.9311386 0.3186277 -2.922 0.003604 **
trend -0.0058320 0.0021985 -2.653 0.008196 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4243 on 605 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 1.042e+06 on 16 and 605 DF, p-value: < 2.2e-16
Estimation results for equation MORT:
=====================================
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.1167328 0.0217427 5.369 1.13e-07 ***
GDP.l1 -0.0002459 0.0002703 -0.910 0.363419
UMP.l1 -0.0026104 0.0251687 -0.104 0.917430
CPI.l1 0.0860609 0.0220777 3.898 0.000108 ***
MORT.l1 1.3691950 0.0431028 31.766 < 2e-16 ***
FFR.l2 -0.1148019 0.0347780 -3.301 0.001020 **
GDP.l2 0.0005009 0.0005481 0.914 0.361074
UMP.l2 0.0236069 0.0375608 0.628 0.529915
CPI.l2 -0.1071027 0.0377620 -2.836 0.004717 **
MORT.l2 -0.7029532 0.0674538 -10.421 < 2e-16 ***
FFR.l3 0.0287427 0.0221629 1.297 0.195165
GDP.l3 -0.0002602 0.0003280 -0.793 0.427917
UMP.l3 -0.0173118 0.0324543 -0.533 0.593939
CPI.l3 0.0240060 0.0227618 1.055 0.292001
MORT.l3 0.2872903 0.0415706 6.911 1.22e-11 ***
const 0.1286240 0.1729351 0.744 0.457304
trend -0.0011842 0.0011933 -0.992 0.321408
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2303 on 605 degrees of freedom
Multiple R-Squared: 0.9952, Adjusted R-squared: 0.9951
F-statistic: 7823 on 16 and 605 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
FFR GDP UMP CPI MORT
FFR 0.22446 1.2807 -0.011102 0.012258 0.045747
GDP 1.28065 1210.0067 -2.213729 0.598938 0.202303
UMP -0.01110 -2.2137 0.122998 -0.008206 -0.003838
CPI 0.01226 0.5989 -0.008206 0.179998 0.008554
MORT 0.04575 0.2023 -0.003838 0.008554 0.053023
Correlation matrix of residuals:
FFR GDP UMP CPI MORT
FFR 1.00000 0.07771 -0.06682 0.06098 0.41933
GDP 0.07771 1.00000 -0.18146 0.04058 0.02526
UMP -0.06682 -0.18146 1.00000 -0.05515 -0.04753
CPI 0.06098 0.04058 -0.05515 1.00000 0.08756
MORT 0.41933 0.02526 -0.04753 0.08756 1.00000VAR(7)
VAR Estimation Results:
=========================
Endogenous variables: FFR, GDP, UMP, CPI, MORT
Deterministic variables: both
Sample size: 618
Log Likelihood: -3688.661
Roots of the characteristic polynomial:
0.9951 0.9951 0.9856 0.9446 0.9446 0.8244 0.8244 0.7781 0.7779 0.7779 0.7754 0.7754 0.7484 0.7484 0.7371 0.7371 0.7304 0.7304 0.7228 0.7228 0.6688 0.6688 0.6663 0.6663 0.6441 0.6441 0.64 0.64 0.6361 0.6166 0.6166 0.5937 0.5937 0.3452 0.0158
Call:
vars::VAR(y = ts_combined, p = 7, type = "both")
Estimation results for equation FFR:
====================================
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 1.403e+00 4.535e-02 30.937 < 2e-16 ***
GDP.l1 8.629e-04 6.405e-04 1.347 0.178400
UMP.l1 -2.535e-02 6.029e-02 -0.420 0.674308
CPI.l1 7.933e-03 4.614e-02 0.172 0.863540
MORT.l1 1.262e-01 9.338e-02 1.352 0.177041
FFR.l2 -5.487e-01 7.527e-02 -7.290 1.02e-12 ***
GDP.l2 -7.138e-04 1.387e-03 -0.515 0.606971
UMP.l2 -9.532e-03 8.487e-02 -0.112 0.910620
CPI.l2 -3.397e-03 8.341e-02 -0.041 0.967528
MORT.l2 -4.680e-01 1.585e-01 -2.951 0.003290 **
FFR.l3 8.124e-02 7.854e-02 1.034 0.301412
GDP.l3 -1.647e-04 1.489e-03 -0.111 0.911958
UMP.l3 -7.866e-02 9.223e-02 -0.853 0.394116
CPI.l3 3.224e-02 8.898e-02 0.362 0.717262
MORT.l3 6.342e-01 1.710e-01 3.707 0.000229 ***
FFR.l4 -1.004e-01 7.847e-02 -1.279 0.201268
GDP.l4 1.490e-04 1.443e-03 0.103 0.917793
UMP.l4 -1.078e-02 9.569e-02 -0.113 0.910357
CPI.l4 -9.002e-02 8.983e-02 -1.002 0.316741
MORT.l4 -3.667e-01 1.743e-01 -2.104 0.035779 *
FFR.l5 1.911e-01 7.844e-02 2.437 0.015124 *
GDP.l5 -5.456e-04 1.414e-03 -0.386 0.699755
UMP.l5 3.382e-02 9.109e-02 0.371 0.710572
CPI.l5 1.196e-01 8.964e-02 1.334 0.182642
MORT.l5 4.568e-02 1.719e-01 0.266 0.790513
FFR.l6 -5.761e-02 7.600e-02 -0.758 0.448764
GDP.l6 -2.237e-05 1.404e-03 -0.016 0.987297
UMP.l6 4.311e-02 9.141e-02 0.472 0.637371
CPI.l6 -8.351e-03 8.485e-02 -0.098 0.921630
MORT.l6 -2.441e-01 1.568e-01 -1.556 0.120146
FFR.l7 -7.134e-03 4.647e-02 -0.153 0.878061
GDP.l7 4.989e-04 7.926e-04 0.630 0.529269
UMP.l7 1.788e-02 7.342e-02 0.243 0.807734
CPI.l7 -6.016e-02 4.819e-02 -1.248 0.212455
MORT.l7 3.022e-01 8.997e-02 3.359 0.000834 ***
const 1.501e-02 3.852e-01 0.039 0.968940
trend -1.314e-03 2.803e-03 -0.469 0.639477
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.468 on 581 degrees of freedom
Multiple R-Squared: 0.987, Adjusted R-squared: 0.9862
F-statistic: 1229 on 36 and 581 DF, p-value: < 2.2e-16
Estimation results for equation GDP:
====================================
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 5.76923 2.95816 1.950 0.05162 .
GDP.l1 1.93459 0.04178 46.307 < 2e-16 ***
UMP.l1 75.04375 3.93310 19.080 < 2e-16 ***
CPI.l1 -2.35828 3.00949 -0.784 0.43358
MORT.l1 4.15780 6.09132 0.683 0.49514
FFR.l2 -6.94597 4.91022 -1.415 0.15772
GDP.l2 -0.72973 0.09046 -8.067 4.16e-15 ***
UMP.l2 -56.83715 5.53646 -10.266 < 2e-16 ***
CPI.l2 2.86289 5.44074 0.526 0.59895
MORT.l2 -5.27329 10.34232 -0.510 0.61033
FFR.l3 -0.28374 5.12349 -0.055 0.95586
GDP.l3 -0.58638 0.09712 -6.037 2.79e-09 ***
UMP.l3 -16.86477 6.01635 -2.803 0.00523 **
CPI.l3 2.68180 5.80457 0.462 0.64424
MORT.l3 -5.46856 11.15763 -0.490 0.62424
FFR.l4 2.02242 5.11884 0.395 0.69292
GDP.l4 0.63387 0.09413 6.734 3.98e-11 ***
UMP.l4 -11.54743 6.24191 -1.850 0.06482 .
CPI.l4 -2.20769 5.85982 -0.377 0.70650
MORT.l4 8.63564 11.36692 0.760 0.44773
FFR.l5 4.36099 5.11679 0.852 0.39440
GDP.l5 -0.23234 0.09223 -2.519 0.01204 *
UMP.l5 14.07096 5.94222 2.368 0.01821 *
CPI.l5 -0.75688 5.84728 -0.129 0.89705
MORT.l5 -11.46292 11.21196 -1.022 0.30703
FFR.l6 -6.59347 4.95742 -1.330 0.18403
GDP.l6 -0.11159 0.09161 -1.218 0.22366
UMP.l6 -3.67265 5.96291 -0.616 0.53819
CPI.l6 1.38198 5.53455 0.250 0.80291
MORT.l6 14.31850 10.23015 1.400 0.16216
FFR.l7 -0.24341 3.03158 -0.080 0.93603
GDP.l7 0.08682 0.05170 1.679 0.09362 .
UMP.l7 1.36762 4.78947 0.286 0.77533
CPI.l7 -1.92011 3.14370 -0.611 0.54158
MORT.l7 -4.16519 5.86874 -0.710 0.47816
const 29.43663 25.12697 1.172 0.24187
trend 0.23645 0.18286 1.293 0.19650
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 30.53 on 581 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 4.472e+05 on 36 and 581 DF, p-value: < 2.2e-16
Estimation results for equation UMP:
====================================
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 -0.0852298 0.0297306 -2.867 0.00430 **
GDP.l1 -0.0016457 0.0004199 -3.919 9.93e-05 ***
UMP.l1 0.6233340 0.0395291 15.769 < 2e-16 ***
CPI.l1 -0.0803553 0.0302465 -2.657 0.00811 **
MORT.l1 0.0280420 0.0612200 0.458 0.64709
FFR.l2 0.1480852 0.0493496 3.001 0.00281 **
GDP.l2 -0.0076606 0.0009092 -8.426 2.82e-16 ***
UMP.l2 -0.0402534 0.0556435 -0.723 0.46972
CPI.l2 0.1115220 0.0546815 2.039 0.04185 *
MORT.l2 -0.0397482 0.1039443 -0.382 0.70231
FFR.l3 -0.0771414 0.0514930 -1.498 0.13465
GDP.l3 0.0148842 0.0009761 15.248 < 2e-16 ***
UMP.l3 0.7253082 0.0604666 11.995 < 2e-16 ***
CPI.l3 -0.1080739 0.0583381 -1.853 0.06445 .
MORT.l3 0.0165723 0.1121384 0.148 0.88256
FFR.l4 0.0077366 0.0514463 0.150 0.88051
GDP.l4 -0.0051300 0.0009460 -5.423 8.63e-08 ***
UMP.l4 -0.3493726 0.0627335 -5.569 3.92e-08 ***
CPI.l4 0.1228308 0.0588934 2.086 0.03745 *
MORT.l4 0.0592520 0.1142418 0.519 0.60420
FFR.l5 -0.0298678 0.0514257 -0.581 0.56160
GDP.l5 -0.0053343 0.0009270 -5.754 1.41e-08 ***
UMP.l5 -0.0161097 0.0597216 -0.270 0.78745
CPI.l5 -0.0740392 0.0587674 -1.260 0.20822
MORT.l5 -0.1108013 0.1126845 -0.983 0.32587
FFR.l6 0.0398457 0.0498240 0.800 0.42419
GDP.l6 0.0077572 0.0009207 8.425 2.83e-16 ***
UMP.l6 0.0355911 0.0599295 0.594 0.55282
CPI.l6 0.0177462 0.0556243 0.319 0.74981
MORT.l6 0.0585746 0.1028169 0.570 0.56910
FFR.l7 0.0110064 0.0304685 0.361 0.71805
GDP.l7 -0.0028968 0.0005196 -5.575 3.79e-08 ***
UMP.l7 0.0050475 0.0481359 0.105 0.91652
CPI.l7 0.0073505 0.0315953 0.233 0.81612
MORT.l7 -0.0190374 0.0589831 -0.323 0.74699
const 0.4379432 0.2525355 1.734 0.08342 .
trend 0.0021652 0.0018378 1.178 0.23923
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3068 on 581 degrees of freedom
Multiple R-Squared: 0.9698, Adjusted R-squared: 0.9679
F-statistic: 517.5 on 36 and 581 DF, p-value: < 2.2e-16
Estimation results for equation CPI:
====================================
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.0361597 0.0407843 0.887 0.37566
GDP.l1 0.0013010 0.0005760 2.259 0.02427 *
UMP.l1 0.1713298 0.0542257 3.160 0.00166 **
CPI.l1 1.4993723 0.0414919 36.136 < 2e-16 ***
MORT.l1 0.1052732 0.0839812 1.254 0.21052
FFR.l2 -0.0094818 0.0676974 -0.140 0.88866
GDP.l2 0.0008479 0.0012472 0.680 0.49688
UMP.l2 -0.0696796 0.0763313 -0.913 0.36170
CPI.l2 -0.6585270 0.0750117 -8.779 < 2e-16 ***
MORT.l2 -0.0903743 0.1425900 -0.634 0.52646
FFR.l3 -0.0218046 0.0706377 -0.309 0.75767
GDP.l3 -0.0021381 0.0013391 -1.597 0.11087
UMP.l3 -0.1751041 0.0829477 -2.111 0.03520 *
CPI.l3 0.2613886 0.0800278 3.266 0.00115 **
MORT.l3 -0.0959855 0.1538307 -0.624 0.53289
FFR.l4 0.0025181 0.0705737 0.036 0.97155
GDP.l4 -0.0009838 0.0012978 -0.758 0.44872
UMP.l4 -0.0582700 0.0860574 -0.677 0.49861
CPI.l4 -0.0115904 0.0807896 -0.143 0.88597
MORT.l4 0.1565024 0.1567161 0.999 0.31839
FFR.l5 0.0343926 0.0705455 0.488 0.62607
GDP.l5 0.0017279 0.0012716 1.359 0.17472
UMP.l5 0.0461154 0.0819257 0.563 0.57372
CPI.l5 -0.1216232 0.0806167 -1.509 0.13193
MORT.l5 -0.2029994 0.1545798 -1.313 0.18962
FFR.l6 0.0028868 0.0683482 0.042 0.96632
GDP.l6 -0.0019113 0.0012630 -1.513 0.13074
UMP.l6 -0.0112452 0.0822109 -0.137 0.89125
CPI.l6 0.0755170 0.0763051 0.990 0.32275
MORT.l6 0.0520858 0.1410435 0.369 0.71205
FFR.l7 -0.0046953 0.0417965 -0.112 0.91059
GDP.l7 0.0012596 0.0007128 1.767 0.07772 .
UMP.l7 0.1198452 0.0660326 1.815 0.07005 .
CPI.l7 -0.0426923 0.0433423 -0.985 0.32503
MORT.l7 0.0524162 0.0809126 0.648 0.51736
const -0.6699990 0.3464265 -1.934 0.05359 .
trend -0.0030076 0.0025211 -1.193 0.23337
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4209 on 581 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 4.618e+05 on 36 and 581 DF, p-value: < 2.2e-16
Estimation results for equation MORT:
=====================================
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 1.170e-01 2.220e-02 5.270 1.93e-07 ***
GDP.l1 -3.297e-04 3.136e-04 -1.051 0.293534
UMP.l1 1.576e-02 2.952e-02 0.534 0.593680
CPI.l1 7.578e-02 2.259e-02 3.355 0.000846 ***
MORT.l1 1.395e+00 4.572e-02 30.508 < 2e-16 ***
FFR.l2 -1.203e-01 3.686e-02 -3.264 0.001163 **
GDP.l2 5.030e-04 6.790e-04 0.741 0.459089
UMP.l2 4.608e-02 4.156e-02 1.109 0.267905
CPI.l2 -1.152e-01 4.084e-02 -2.821 0.004951 **
MORT.l2 -7.614e-01 7.763e-02 -9.808 < 2e-16 ***
FFR.l3 6.682e-02 3.846e-02 1.738 0.082818 .
GDP.l3 5.567e-04 7.290e-04 0.764 0.445411
UMP.l3 -3.250e-02 4.516e-02 -0.720 0.471965
CPI.l3 6.360e-02 4.357e-02 1.460 0.144904
MORT.l3 3.656e-01 8.375e-02 4.366 1.50e-05 ***
FFR.l4 -8.747e-02 3.842e-02 -2.277 0.023169 *
GDP.l4 -5.624e-04 7.065e-04 -0.796 0.426325
UMP.l4 -5.438e-02 4.685e-02 -1.161 0.246269
CPI.l4 1.744e-02 4.398e-02 0.396 0.691923
MORT.l4 -1.167e-02 8.532e-02 -0.137 0.891287
FFR.l5 1.047e-01 3.841e-02 2.726 0.006608 **
GDP.l5 -7.861e-04 6.923e-04 -1.135 0.256669
UMP.l5 -4.963e-02 4.460e-02 -1.113 0.266314
CPI.l5 -3.875e-02 4.389e-02 -0.883 0.377673
MORT.l5 -7.099e-02 8.416e-02 -0.844 0.399248
FFR.l6 -5.433e-02 3.721e-02 -1.460 0.144835
GDP.l6 4.214e-04 6.876e-04 0.613 0.540226
UMP.l6 3.897e-02 4.476e-02 0.871 0.384305
CPI.l6 1.259e-02 4.154e-02 0.303 0.762026
MORT.l6 -8.260e-04 7.679e-02 -0.011 0.991421
FFR.l7 -9.717e-05 2.275e-02 -0.004 0.996594
GDP.l7 1.814e-04 3.881e-04 0.467 0.640338
UMP.l7 4.285e-02 3.595e-02 1.192 0.233739
CPI.l7 -1.500e-02 2.360e-02 -0.636 0.525271
MORT.l7 4.212e-02 4.405e-02 0.956 0.339370
const 2.204e-01 1.886e-01 1.169 0.243072
trend 4.001e-05 1.373e-03 0.029 0.976756
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2291 on 581 degrees of freedom
Multiple R-Squared: 0.9954, Adjusted R-squared: 0.9951
F-statistic: 3512 on 36 and 581 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
FFR GDP UMP CPI MORT
FFR 0.219033 1.3061 -0.010856 0.008961 0.044674
GDP 1.306052 932.0092 -0.995366 0.353180 0.194479
UMP -0.010856 -0.9954 0.094142 -0.002369 -0.004714
CPI 0.008961 0.3532 -0.002369 0.177159 0.006045
MORT 0.044674 0.1945 -0.004714 0.006045 0.052507
Correlation matrix of residuals:
FFR GDP UMP CPI MORT
FFR 1.00000 0.09141 -0.07560 0.04549 0.41657
GDP 0.09141 1.00000 -0.10626 0.02749 0.02780
UMP -0.07560 -0.10626 1.00000 -0.01834 -0.06705
CPI 0.04549 0.02749 -0.01834 1.00000 0.06267
MORT 0.41657 0.02780 -0.06705 0.06267 1.00000VAR(10)
VAR Estimation Results:
=========================
Endogenous variables: FFR, GDP, UMP, CPI, MORT
Deterministic variables: both
Sample size: 615
Log Likelihood: -3563.255
Roots of the characteristic polynomial:
0.9918 0.9918 0.9786 0.9494 0.9494 0.8982 0.8705 0.8705 0.8669 0.8669 0.8619 0.8619 0.8344 0.8344 0.8339 0.8339 0.8286 0.8286 0.8263 0.8263 0.8261 0.8138 0.8138 0.8059 0.8059 0.8032 0.8032 0.7879 0.7879 0.787 0.787 0.781 0.781 0.7606 0.7606 0.7562 0.7562 0.7546 0.7546 0.7481 0.7481 0.7469 0.7469 0.7386 0.7118 0.7118 0.6778 0.6778 0.6435 0.6435
Call:
vars::VAR(y = ts_combined, p = 10, type = "both")
Estimation results for equation FFR:
====================================
FFR = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 1.3925124 0.0456139 30.528 < 2e-16 ***
GDP.l1 0.0006535 0.0006170 1.059 0.290016
UMP.l1 -0.0249535 0.0631346 -0.395 0.692814
CPI.l1 0.0146835 0.0443089 0.331 0.740474
MORT.l1 0.1577118 0.0903379 1.746 0.081391 .
FFR.l2 -0.5144433 0.0756368 -6.801 2.65e-11 ***
GDP.l2 -0.0002373 0.0013374 -0.177 0.859261
UMP.l2 0.0508775 0.0889707 0.572 0.567655
CPI.l2 -0.0097948 0.0801393 -0.122 0.902767
MORT.l2 -0.4677727 0.1539358 -3.039 0.002486 **
FFR.l3 0.0750617 0.0783067 0.959 0.338192
GDP.l3 -0.0006841 0.0015109 -0.453 0.650883
UMP.l3 -0.1413837 0.0945131 -1.496 0.135236
CPI.l3 0.0260103 0.0860013 0.302 0.762427
MORT.l3 0.4901616 0.1659450 2.954 0.003270 **
FFR.l4 -0.1161904 0.0765430 -1.518 0.129582
GDP.l4 0.0009027 0.0018361 0.492 0.623173
UMP.l4 0.0293343 0.1021184 0.287 0.774021
CPI.l4 -0.0865638 0.0868835 -0.996 0.319523
MORT.l4 -0.2423112 0.1678998 -1.443 0.149523
FFR.l5 0.2098276 0.0760056 2.761 0.005956 **
GDP.l5 -0.0016801 0.0019176 -0.876 0.381324
UMP.l5 0.0425428 0.1039950 0.409 0.682633
CPI.l5 0.1195584 0.0873215 1.369 0.171490
MORT.l5 0.0877093 0.1685299 0.520 0.602963
FFR.l6 0.0125900 0.0763905 0.165 0.869152
GDP.l6 0.0009697 0.0018917 0.513 0.608412
UMP.l6 0.0681907 0.1031392 0.661 0.508785
CPI.l6 -0.0549973 0.0874025 -0.629 0.529446
MORT.l6 -0.4285943 0.1685841 -2.542 0.011279 *
FFR.l7 -0.2918106 0.0763471 -3.822 0.000147 ***
GDP.l7 0.0015311 0.0016305 0.939 0.348116
UMP.l7 -0.0058253 0.0952556 -0.061 0.951258
CPI.l7 0.0544967 0.0871326 0.625 0.531932
MORT.l7 0.8164602 0.1696519 4.813 1.92e-06 ***
FFR.l8 0.3174611 0.0772681 4.109 4.57e-05 ***
GDP.l8 -0.0023407 0.0015428 -1.517 0.129791
UMP.l8 -0.0980588 0.0880199 -1.114 0.265732
CPI.l8 -0.1037529 0.0863820 -1.201 0.230220
MORT.l8 -0.3277054 0.1704243 -1.923 0.054999 .
FFR.l9 -0.0539941 0.0765108 -0.706 0.480663
GDP.l9 0.0005875 0.0014808 0.397 0.691708
UMP.l9 0.0037559 0.0880419 0.043 0.965987
CPI.l9 0.0725219 0.0813894 0.891 0.373284
MORT.l9 0.0281445 0.1594137 0.177 0.859926
FFR.l10 -0.0880616 0.0462636 -1.903 0.057489 .
GDP.l10 0.0003370 0.0007986 0.422 0.673197
UMP.l10 0.0550137 0.0708891 0.776 0.438043
CPI.l10 -0.0341610 0.0465982 -0.733 0.463805
MORT.l10 -0.0768781 0.0912497 -0.843 0.399865
const 0.1470506 0.3953724 0.372 0.710085
trend -0.0008793 0.0030032 -0.293 0.769798
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4456 on 563 degrees of freedom
Multiple R-Squared: 0.9886, Adjusted R-squared: 0.9876
F-statistic: 958.8 on 51 and 563 DF, p-value: < 2.2e-16
Estimation results for equation GDP:
====================================
GDP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 4.98895 3.11725 1.600 0.11006
GDP.l1 1.93129 0.04217 45.799 < 2e-16 ***
UMP.l1 73.04305 4.31460 16.929 < 2e-16 ***
CPI.l1 -2.46602 3.02806 -0.814 0.41577
MORT.l1 4.06364 6.17367 0.658 0.51067
FFR.l2 -4.81802 5.16901 -0.932 0.35169
GDP.l2 -0.71996 0.09140 -7.877 1.74e-14 ***
UMP.l2 -52.37044 6.08024 -8.613 < 2e-16 ***
CPI.l2 3.24432 5.47671 0.592 0.55383
MORT.l2 -3.58085 10.51995 -0.340 0.73369
FFR.l3 -0.99520 5.35147 -0.186 0.85254
GDP.l3 -0.64714 0.10325 -6.268 7.30e-10 ***
UMP.l3 -17.53285 6.45901 -2.714 0.00684 **
CPI.l3 2.60111 5.87731 0.443 0.65825
MORT.l3 -9.74289 11.34065 -0.859 0.39064
FFR.l4 2.09603 5.23094 0.401 0.68879
GDP.l4 0.76221 0.12548 6.075 2.29e-09 ***
UMP.l4 -9.31023 6.97875 -1.334 0.18272
CPI.l4 -2.71479 5.93760 -0.457 0.64769
MORT.l4 9.10700 11.47424 0.794 0.42771
FFR.l5 4.23222 5.19421 0.815 0.41553
GDP.l5 -0.29789 0.13105 -2.273 0.02339 *
UMP.l5 12.54793 7.10700 1.766 0.07801 .
CPI.l5 0.17272 5.96753 0.029 0.97692
MORT.l5 -6.16596 11.51730 -0.535 0.59261
FFR.l6 -6.44129 5.22051 -1.234 0.21778
GDP.l6 -0.19456 0.12928 -1.505 0.13290
UMP.l6 -4.13621 7.04851 -0.587 0.55756
CPI.l6 -0.75880 5.97307 -0.127 0.89896
MORT.l6 7.27257 11.52100 0.631 0.52814
FFR.l7 -0.47794 5.21755 -0.092 0.92705
GDP.l7 0.27372 0.11143 2.457 0.01433 *
UMP.l7 2.63440 6.50975 0.405 0.68586
CPI.l7 0.69867 5.95462 0.117 0.90664
MORT.l7 3.90808 11.59398 0.337 0.73618
FFR.l8 -0.13376 5.28049 -0.025 0.97980
GDP.l8 -0.11138 0.10544 -1.056 0.29124
UMP.l8 -3.52894 6.01526 -0.587 0.55766
CPI.l8 4.16021 5.90333 0.705 0.48127
MORT.l8 -0.25103 11.64676 -0.022 0.98281
FFR.l9 4.82701 5.22873 0.923 0.35631
GDP.l9 -0.05080 0.10120 -0.502 0.61587
UMP.l9 -2.09553 6.01676 -0.348 0.72776
CPI.l9 -9.39502 5.56214 -1.689 0.09175 .
MORT.l9 -6.02749 10.89430 -0.553 0.58030
FFR.l10 -6.09321 3.16165 -1.927 0.05445 .
GDP.l10 0.04899 0.05457 0.898 0.36969
UMP.l10 2.40878 4.84455 0.497 0.61923
CPI.l10 4.06839 3.18451 1.278 0.20193
MORT.l10 3.11147 6.23598 0.499 0.61801
const 30.62362 27.01967 1.133 0.25754
trend 0.28405 0.20524 1.384 0.16690
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 30.45 on 563 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 3.139e+05 on 51 and 563 DF, p-value: < 2.2e-16
Estimation results for equation UMP:
====================================
UMP = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 -0.0962662 0.0295834 -3.254 0.001206 **
GDP.l1 -0.0015077 0.0004002 -3.768 0.000182 ***
UMP.l1 0.6983075 0.0409466 17.054 < 2e-16 ***
CPI.l1 -0.0807023 0.0287370 -2.808 0.005153 **
MORT.l1 0.0190020 0.0585896 0.324 0.745813
FFR.l2 0.1580754 0.0490551 3.222 0.001345 **
GDP.l2 -0.0082853 0.0008674 -9.552 < 2e-16 ***
UMP.l2 -0.0554728 0.0577029 -0.961 0.336788
CPI.l2 0.1157673 0.0519752 2.227 0.026318 *
MORT.l2 -0.0112406 0.0998367 -0.113 0.910396
FFR.l3 -0.0778622 0.0507867 -1.533 0.125808
GDP.l3 0.0172790 0.0009799 17.634 < 2e-16 ***
UMP.l3 0.6025063 0.0612975 9.829 < 2e-16 ***
CPI.l3 -0.0980851 0.0557770 -1.759 0.079202 .
MORT.l3 -0.0077874 0.1076254 -0.072 0.942344
FFR.l4 -0.0072534 0.0496428 -0.146 0.883886
GDP.l4 -0.0072583 0.0011908 -6.095 2.03e-09 ***
UMP.l4 -0.3539175 0.0662300 -5.344 1.32e-07 ***
CPI.l4 0.0905868 0.0563493 1.608 0.108484
MORT.l4 0.0794237 0.1088932 0.729 0.466077
FFR.l5 -0.0048509 0.0492942 -0.098 0.921644
GDP.l5 -0.0087046 0.0012437 -6.999 7.36e-12 ***
UMP.l5 -0.0701354 0.0674471 -1.040 0.298852
CPI.l5 -0.0427311 0.0566333 -0.755 0.450851
MORT.l5 -0.1564678 0.1093019 -1.432 0.152836
FFR.l6 0.0176201 0.0495439 0.356 0.722239
GDP.l6 0.0147355 0.0012269 12.010 < 2e-16 ***
UMP.l6 0.2278423 0.0668921 3.406 0.000706 ***
CPI.l6 0.0137721 0.0566858 0.243 0.808128
MORT.l6 0.1375262 0.1093370 1.258 0.208979
FFR.l7 0.0779449 0.0495158 1.574 0.116016
GDP.l7 -0.0066642 0.0010575 -6.302 5.93e-10 ***
UMP.l7 -0.0675298 0.0617791 -1.093 0.274824
CPI.l7 -0.0141871 0.0565108 -0.251 0.801866
MORT.l7 -0.1480836 0.1100296 -1.346 0.178892
FFR.l8 -0.1522983 0.0501130 -3.039 0.002483 **
GDP.l8 -0.0030433 0.0010006 -3.041 0.002465 **
UMP.l8 0.0157603 0.0570862 0.276 0.782589
CPI.l8 0.0040020 0.0560240 0.071 0.943079
MORT.l8 0.1079815 0.1105305 0.977 0.329019
FFR.l9 0.1189025 0.0496219 2.396 0.016893 *
GDP.l9 0.0061626 0.0009604 6.417 2.96e-10 ***
UMP.l9 0.0614381 0.0571005 1.076 0.282404
CPI.l9 0.0246464 0.0527860 0.467 0.640744
MORT.l9 0.0333963 0.1033895 0.323 0.746804
FFR.l10 -0.0188796 0.0300048 -0.629 0.529460
GDP.l10 -0.0027436 0.0005179 -5.298 1.69e-07 ***
UMP.l10 -0.0771065 0.0459759 -1.677 0.094076 .
CPI.l10 -0.0166136 0.0302217 -0.550 0.582727
MORT.l10 -0.0627474 0.0591810 -1.060 0.289479
const 0.4920408 0.2564230 1.919 0.055507 .
trend 0.0024495 0.0019477 1.258 0.209052
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.289 on 563 degrees of freedom
Multiple R-Squared: 0.974, Adjusted R-squared: 0.9716
F-statistic: 413.6 on 51 and 563 DF, p-value: < 2.2e-16
Estimation results for equation CPI:
====================================
CPI = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 3.907e-02 4.331e-02 0.902 0.36735
GDP.l1 1.294e-03 5.859e-04 2.208 0.02763 *
UMP.l1 1.600e-01 5.994e-02 2.669 0.00782 **
CPI.l1 1.502e+00 4.207e-02 35.703 < 2e-16 ***
MORT.l1 1.294e-01 8.577e-02 1.508 0.13203
FFR.l2 -3.812e-02 7.181e-02 -0.531 0.59574
GDP.l2 8.255e-04 1.270e-03 0.650 0.51589
UMP.l2 -2.796e-02 8.447e-02 -0.331 0.74074
CPI.l2 -6.645e-01 7.609e-02 -8.734 < 2e-16 ***
MORT.l2 -1.235e-01 1.462e-01 -0.845 0.39847
FFR.l3 2.957e-02 7.435e-02 0.398 0.69098
GDP.l3 -2.246e-03 1.435e-03 -1.566 0.11797
UMP.l3 -2.336e-01 8.974e-02 -2.603 0.00948 **
CPI.l3 2.646e-01 8.166e-02 3.240 0.00127 **
MORT.l3 -1.097e-01 1.576e-01 -0.696 0.48651
FFR.l4 -2.956e-02 7.268e-02 -0.407 0.68435
GDP.l4 -2.533e-04 1.743e-03 -0.145 0.88451
UMP.l4 -4.678e-03 9.696e-02 -0.048 0.96154
CPI.l4 -2.182e-02 8.249e-02 -0.264 0.79149
MORT.l4 1.778e-01 1.594e-01 1.115 0.26524
FFR.l5 4.774e-02 7.216e-02 0.662 0.50849
GDP.l5 9.014e-05 1.821e-03 0.050 0.96053
UMP.l5 9.037e-03 9.874e-02 0.092 0.92711
CPI.l5 -1.034e-01 8.291e-02 -1.247 0.21287
MORT.l5 -1.870e-01 1.600e-01 -1.168 0.24313
FFR.l6 -5.537e-04 7.253e-02 -0.008 0.99391
GDP.l6 -2.867e-04 1.796e-03 -0.160 0.87326
UMP.l6 6.769e-02 9.793e-02 0.691 0.48974
CPI.l6 2.153e-02 8.299e-02 0.259 0.79536
MORT.l6 -1.278e-02 1.601e-01 -0.080 0.93637
FFR.l7 -3.751e-02 7.249e-02 -0.517 0.60503
GDP.l7 1.515e-03 1.548e-03 0.979 0.32807
UMP.l7 1.047e-01 9.044e-02 1.158 0.24737
CPI.l7 8.547e-02 8.273e-02 1.033 0.30200
MORT.l7 2.285e-01 1.611e-01 1.418 0.15664
FFR.l8 7.617e-02 7.336e-02 1.038 0.29962
GDP.l8 -1.447e-03 1.465e-03 -0.988 0.32369
UMP.l8 -5.184e-02 8.357e-02 -0.620 0.53529
CPI.l8 -9.164e-02 8.202e-02 -1.117 0.26435
MORT.l8 -2.547e-01 1.618e-01 -1.574 0.11607
FFR.l9 -1.029e-01 7.264e-02 -1.416 0.15720
GDP.l9 7.744e-04 1.406e-03 0.551 0.58201
UMP.l9 -5.111e-02 8.359e-02 -0.611 0.54117
CPI.l9 2.656e-02 7.728e-02 0.344 0.73115
MORT.l9 2.816e-01 1.514e-01 1.860 0.06334 .
FFR.l10 5.607e-02 4.393e-02 1.277 0.20228
GDP.l10 -1.880e-04 7.582e-04 -0.248 0.80422
UMP.l10 5.368e-02 6.731e-02 0.798 0.42545
CPI.l10 -1.938e-02 4.424e-02 -0.438 0.66150
MORT.l10 -1.574e-01 8.664e-02 -1.817 0.06978 .
const -4.635e-01 3.754e-01 -1.235 0.21746
trend -1.469e-03 2.851e-03 -0.515 0.60670
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.423 on 563 degrees of freedom
Multiple R-Squared: 1, Adjusted R-squared: 1
F-statistic: 3.18e+05 on 51 and 563 DF, p-value: < 2.2e-16
Estimation results for equation MORT:
=====================================
MORT = FFR.l1 + GDP.l1 + UMP.l1 + CPI.l1 + MORT.l1 + FFR.l2 + GDP.l2 + UMP.l2 + CPI.l2 + MORT.l2 + FFR.l3 + GDP.l3 + UMP.l3 + CPI.l3 + MORT.l3 + FFR.l4 + GDP.l4 + UMP.l4 + CPI.l4 + MORT.l4 + FFR.l5 + GDP.l5 + UMP.l5 + CPI.l5 + MORT.l5 + FFR.l6 + GDP.l6 + UMP.l6 + CPI.l6 + MORT.l6 + FFR.l7 + GDP.l7 + UMP.l7 + CPI.l7 + MORT.l7 + FFR.l8 + GDP.l8 + UMP.l8 + CPI.l8 + MORT.l8 + FFR.l9 + GDP.l9 + UMP.l9 + CPI.l9 + MORT.l9 + FFR.l10 + GDP.l10 + UMP.l10 + CPI.l10 + MORT.l10 + const + trend
Estimate Std. Error t value Pr(>|t|)
FFR.l1 0.1328152 0.0231328 5.741 1.54e-08 ***
GDP.l1 -0.0003791 0.0003129 -1.211 0.226282
UMP.l1 0.0048796 0.0320183 0.152 0.878926
CPI.l1 0.0766326 0.0224710 3.410 0.000695 ***
MORT.l1 1.4076295 0.0458143 30.725 < 2e-16 ***
FFR.l2 -0.1419178 0.0383587 -3.700 0.000237 ***
GDP.l2 0.0006852 0.0006783 1.010 0.312795
UMP.l2 0.0590239 0.0451209 1.308 0.191364
CPI.l2 -0.1182545 0.0406421 -2.910 0.003761 **
MORT.l2 -0.7692118 0.0780676 -9.853 < 2e-16 ***
FFR.l3 0.0646591 0.0397128 1.628 0.104048
GDP.l3 0.0001740 0.0007662 0.227 0.820449
UMP.l3 -0.0530320 0.0479317 -1.106 0.269023
CPI.l3 0.0596652 0.0436150 1.368 0.171859
MORT.l3 0.3466754 0.0841579 4.119 4.37e-05 ***
FFR.l4 -0.0839070 0.0388183 -2.162 0.031075 *
GDP.l4 -0.0001326 0.0009311 -0.142 0.886812
UMP.l4 -0.0368019 0.0517887 -0.711 0.477616
CPI.l4 0.0238755 0.0440624 0.542 0.588132
MORT.l4 0.0234867 0.0851493 0.276 0.782780
FFR.l5 0.1093545 0.0385457 2.837 0.004718 **
GDP.l5 -0.0011464 0.0009725 -1.179 0.238969
UMP.l5 -0.0633593 0.0527404 -1.201 0.230123
CPI.l5 -0.0450105 0.0442845 -1.016 0.309879
MORT.l5 -0.0921020 0.0854689 -1.078 0.281670
FFR.l6 -0.0200262 0.0387409 -0.517 0.605410
GDP.l6 0.0005742 0.0009594 0.598 0.549767
UMP.l6 0.0645909 0.0523064 1.235 0.217399
CPI.l6 0.0091826 0.0443256 0.207 0.835958
MORT.l6 -0.0283568 0.0854963 -0.332 0.740260
FFR.l7 -0.1422721 0.0387190 -3.674 0.000261 ***
GDP.l7 0.0002795 0.0008269 0.338 0.735510
UMP.l7 0.0160036 0.0483083 0.331 0.740556
CPI.l7 0.0027166 0.0441887 0.061 0.951001
MORT.l7 0.2056750 0.0860379 2.391 0.017152 *
FFR.l8 0.1732977 0.0391860 4.422 1.17e-05 ***
GDP.l8 0.0002608 0.0007824 0.333 0.739011
UMP.l8 0.0304853 0.0446387 0.683 0.494930
CPI.l8 -0.0318458 0.0438081 -0.727 0.467566
MORT.l8 -0.1808582 0.0864296 -2.093 0.036836 *
FFR.l9 -0.0849790 0.0388020 -2.190 0.028929 *
GDP.l9 -0.0008728 0.0007510 -1.162 0.245666
UMP.l9 -0.0730508 0.0446499 -1.636 0.102381
CPI.l9 0.0678871 0.0412761 1.645 0.100589
MORT.l9 0.0095171 0.0808456 0.118 0.906332
FFR.l10 0.0232036 0.0234623 0.989 0.323100
GDP.l10 0.0005296 0.0004050 1.308 0.191500
UMP.l10 0.0599008 0.0359509 1.666 0.096234 .
CPI.l10 -0.0459623 0.0236320 -1.945 0.052282 .
MORT.l10 0.0302584 0.0462767 0.654 0.513470
const 0.3377335 0.2005106 1.684 0.092665 .
trend 0.0009649 0.0015230 0.634 0.526659
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.226 on 563 degrees of freedom
Multiple R-Squared: 0.9957, Adjusted R-squared: 0.9953
F-statistic: 2550 on 51 and 563 DF, p-value: < 2.2e-16
Covariance matrix of residuals:
FFR GDP UMP CPI MORT
FFR 0.198526 0.8625 -0.007206 0.004755 0.040735
GDP 0.862483 927.1797 -0.745157 0.322888 0.201382
UMP -0.007206 -0.7452 0.083506 -0.003017 -0.003125
CPI 0.004755 0.3229 -0.003017 0.178968 0.004934
MORT 0.040735 0.2014 -0.003125 0.004934 0.051060
Correlation matrix of residuals:
FFR GDP UMP CPI MORT
FFR 1.00000 0.06357 -0.05596 0.02523 0.40460
GDP 0.06357 1.00000 -0.08469 0.02507 0.02927
UMP -0.05596 -0.08469 1.00000 -0.02468 -0.04785
CPI 0.02523 0.02507 -0.02468 1.00000 0.05161
MORT 0.40460 0.02927 -0.04785 0.05161 1.00000Hyperparameter Optimization
Model fitting with minimum AIC:
10, 7646.50982266218, 8796.13161229187
Model fitting with minimum BIC:
3, 8044.25621302432, 8421.05612090715Cross Validation
The plots compare the cross-validation RMSE for VAR(3) and VAR(10) models across different financial variables over time. The consistently lower RMSE values for VAR(1) indicate that it provides a better fit or more accurate forecasts for these series.
Forecasting
The VAR(3) model forecasts for key economic indicators suggest a tightening monetary environment, with rising Federal Funds Rates potentially in response to upward inflationary trends indicated by the Consumer Price Index. Consistent economic growth is anticipated, as evidenced by the steady increase in GDP, though the projection comes with increasing uncertainty. The sharp uptick in forecasted 30-Year Mortgage Rates could dampen housing market activity by raising borrowing costs. Meanwhile, the expected rise in the unemployment rate signals possible headwinds for the labor market, which may require targeted economic policies to mitigate the risk of a slowdown and support job growth.
